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Un ive rs i ty o f K on s ta n z De pa rtmen t o f Econ om ics

Public Debt and Total Factor Productivity

Leo Kaas

Working Paper Series 2014-22

Public Debt and Total Factor Productivity∗

Leo Kaas†

December 2014

Abstract

This paper explores the role of public debt and fiscal deficits on factor productivity

in an economy with credit market frictions and heterogeneous firms. When credit

market conditions are sufficiently weak, low interest rates permit the government to

run Ponzi schemes so that permanent primary deficits can be sustained. For small

enough deficit ratios, the model has two steady states of which one is an unstable

bubble and the other one is stable. The stable equilibrium features higher levels of

credit and capital, but also a lower interest rate, lower total factor productivity and

output. The model is calibrated to the US economy to derive the maximum sustain-

able deficit ratio and to examine the dynamic responses to changes in debt policy.

A reduction of the primary deficit triggers an expansion of credit and capital, but it

also leads to a deterioration of total factor productivity since more low-productivity

firms prefer to remain active at the lower equilibrium interest rate.

JEL classification: D92, E62, H62

Keywords: Credit constraints; Unbacked public debt; Dynamic inefficiency; Sustainable

deficits

∗I am grateful to Almuth Scholl for helpful comments. I also thank the German Research Foundation

(grant No. KA 1519/4) for financial support. The usual disclaimer applies.†University of Konstanz, E-mail: leo.kaas@uni-konstanz.de

1 Introduction

Large fiscal deficits and rising stocks of public debt in many advanced economies, especially

in the aftermath of the Great Recession, raise concerns about debt sustainability and

about adverse consequences for the private sector, in particular for credit supply and

investment. Empirical studies point at a non-monotonic relationship between public debt

and economic growth (see e.g. Kumar and Woo (2010), Reinhart and Rogoff (2010) and

Baum et al. (2013)). Examining the channels through which debt impacts growth, Pattillo

et al. (2011) find for a sample of 93 developing countries that the majority of the effect

of public debt on output growth occurs via total factor productivity (TFP) rather than

via capital accumulation. On the other hand, theoretical and quantitative studies on debt

policy (see the literature surveyed below) traditionally focus on the effects of public debt on

capital investment and interest rates, while treating the dynamics of TFP as an exogenous

process.

To fill this gap, this paper examines the effect of public debt and fiscal deficits on TFP,

using a tractable dynamic general equilibrium model with credit market imperfections

and heterogeneous firms facing idiosyncratic productivity shocks. Due to binding collat-

eral constraints, the capital allocation among firms, and therefore TFP, are endogenous

variables. If the credit constraints are sufficiently tight, low interest rates permit the gov-

ernment to run Ponzi schemes, i.e. to roll over unbacked government debt indefinitely,

so that permanent primary deficits can be sustained. Such economies typically have two

steady states. One steady state has a larger stock of public debt, a higher interest rate

and a more efficient factor allocation, but at the same time less private credit and capital.

This steady state is generally unstable and can only be sustained if the price of government

bonds is a bubble. In the absence of such a bubble, the economy converges to the other,

stable steady state at which TFP and the interest rate are lower, while private credit and

capital are larger.

After describing the model and deriving the key dynamic relationships in Section 2, I prove

in Section 3 several properties about the different stationary equilibria. In particular, I

derive conditions for existence and uniqueness and show that TFP is always increasing if

credit market conditions improve, regardless of the existence of a bubble. Further, it is

shown that primary deficits up to a maximal level can be sustained whenever the economy

is dynamically inefficient, i.e. when its interest rate is smaller than the growth rate in the

absence of a bubble.

To determine the maximum sustainable deficit ratio and to address further quantitative

questions, I calibrate the model in Section 4 to match long-run features of the US econ-

1

omy. It turns out that the maximum sustainable primary deficit is actually quite small,

amounting to less than one percent of output. This number is much lower than the 5.2%

threshold that Chalk (2000) obtains for the maximal deficit ratio based on a quantitative

overlapping generations model.

Another finding for the calibrated model is that changes of the deficit ratio have opposite

effects on credit and capital, on the one hand, and on TFP, on the other hand. A reduction

of the deficit, for example, reduces the interest rate which triggers a surge in private credit

and capital investment. At the same time, however, the lower interest rate makes it more

attractive for low-productivity firms to stay in business which has a detrimental effect on

aggregate productivity. I show that the second effect dominates the overall impact on

aggregate output which necessarily falls in response to the deficit reduction in the very

long run. Yet, this long run effect only materializes after several decades and is offset by

substantial output gains in the medium run. A further quantitative finding concerns the

effect of changes in financial conditions. It turns out that even significant expansions of

private credit induced by a relaxation of collateral constraints have quite modest effects on

aggregate output and TFP. For example, a 10 percent expansion of firm credit merely raises

TFP by 0.3 percent and output by 0.5 percent. This contrasts with some of the recent

business-cycle literature that finds large output responses to financial shocks (e.g. Jermann

and Quadrini (2012)).

This paper relates to the literature on overlapping generations models in which dynamic

inefficiency can give rise to the existence of rational bubbles and to the possibility of debt

Ponzi schemes so that governments are able to finance primary deficits indefinitely (Di-

amond (1965), Tirole (1985), Blanchard (1985) and Weil (1987)).1 Bullard and Russell

(1999) and Chalk (2000) consider quantitative overlapping generations economies with low

interest rates and unbacked debt, and the latter explicitly calculates maximum sustainable

deficit levels. Rankin and Roffia (2003) are interested in maximum sustainable debt (rather

than deficits) in an overlapping generations model. In all these papers, however, TFP is

exogenous. The same is true in the literature on incomplete markets with infinitely-lived

households facing idiosyncratic income risk where a dynamically inefficient overaccumula-

tion of capital is possible, so that unbacked public debt can be sustained (e.g. Aiyagari

and McGrattan (1998)).

The paper also relates to the literature on rational bubbles in economies with infinitely-lived

agents and credit market frictions (e.g. Kocherlakota (2008) and Hellwig and Lorenzoni

1The tight connection between dynamic inefficiency and the possibility of Ponzi schemes is only valid

in the absence of aggregate uncertainty (see e.g. Blanchard and Weil (2001)).

2

(2009)). As in this paper, Kocherlakota (2009) and Miao and Wang (2012, 2013) argue

that rational bubbles can improve production efficiency when credit constraints are tight.

The model of this paper builds upon Azariadis et al. (2014) who show that the dynamics

of unsecured credit, which responds to changes in self-fulfilling beliefs, affects aggregate

factor productivity. Different from these contributions, this paper explores the impact of

unbacked government debt and fiscal deficits on TFP.

2 The Model

2.1 The Environment

Consider a general equilibrium model in discrete time with a representative worker house-

hold and a continuum i ∈ [0, 1] of firms, each owned by a single shareholder. The govern-

ment runs primary surpluses or deficits and issues public debt. Firms face idiosyncratic

productivity shocks. Their owners can invest either in their own business, which is optimal

if productivity is sufficiently high, or in financial assets. There is an asset market for busi-

ness credit which operates in each period, after firm productivities are revealed. Besides

lending to other firms, firm owners can also hold government bonds. While government

liabilities can principally be interpreted as either bonds or fiat money, this paper favors the

first interpretation. This is because fiscal governments in modern economies finance their

deficits primarily by issuing bonds rather than by seignorage. Moreover, in this model

money has no essential role that is separate from the other assets.

At any time t, firm owner i maximizes expected discounted utility

Et∑

τ≥t

βτ−t ln(C iτ ) (1)

over future consumption streams. Firm i has access to a production technology which

produces the single consumption/investment good Y it = F (zitK

it , AtL

it) from inputs of

capital Kit and labor Lit. F is a constant-returns and concave production function. zit is

an idiosyncratic capital productivity shock, which is realized in period t − 1 when firms

decide about capital investment.2 These idiosyncratic productivities are distributed with

continuous cumulative distribution function G(z) and they are uncorrelated across time.

The labor efficiency parameter At grows with constant rate g ≥ 0. Capital depreciates

during the period at rate δ.

2The timing notation follows the convention in the macroeconomic literature in that capital employed

at date t carries index t, although it is decided at time t− 1. Idiosyncratic productivity therefore has the

same time index.

3

The market for business credit operates in period t after idiosyncratic productivities are

revealed. Owners of more productive firms borrow from the owners of less productive firms

at gross interest rate Rt+1.3 As borrowers cannot promise to repay their debts in the next

period t+1, they need to provide collateral to their lenders. I assume that a fixed fraction

λ ∈ (0, 1] of the firm’s capital stock provides secure collateral. Thus, when Bit+1 denotes

the firm owner’s debt issued in period t, the collateral constraint4 in period t reads as

Rt+1Bit+1 ≤ λ(1− δ)Ki

t+1 . (2)

Owners of low-productivity firms optimally decide to not invest in their own businesses

but rather to hold financial assets. In period t, these firm owners supply −Bit+1 > 0 units

of credit to other firms or buy Dit+1 units of government bonds at price qt. The budget

constraint of firm owner i in period t reads as follows:

C it +Ki

t+1 −Bit+1 + qtD

it+1 = Y i

t − wtLit + (1− δ)Ki

t − RtBit + qtD

it . (3)

The right-hand side of this budget constraint represents the owner’s wealth in period t.

Labor is hired at real wage wt to produce output Y it , the firm owner repays credit (or is

repaid) and holds the undepreciated stock of capital and possibly some government bonds.

The firm owner can spend his wealth on consumption, on investment in his own business,

with equity Kit+1 − Bi

t+1, or on public bonds.

The aggregate supply of government bonds Dt changes over time when the government

runs deficits or surpluses. The government budget constraint in period t is

qt(Dt+1 −Dt) = Xt , (4)

where Xt is the primary deficit (or primary surplus, when negative). To simplify the role of

the government, it is assumed that deficits (surpluses) are transferred to (collected from)

the worker household in a lump-sum fashion.

The worker household supplies labor and is not active in asset markets, thus consumes

labor income net of taxes/transfers in every period. The absence of the worker household

from asset markets is not a severe restriction. Indeed, if the worker household has the same

preferences as firm owners, he would decide not to save in any steady state equilibrium of

3One can think of this credit as including corporate bonds (directly held by the lender) and loans that

are granted through financial intermediaries (which are not explicitly modeled).4Loss of collateral is the only punishment of a defaulting owner. Azariadis et al. (2014) additionally

allow for unsecured credit that rests on the borrower’s reputation which is harmed in a default event.

Neither setting has default in equilibrium because productivities are revealed before credit is exchanged.

Cui and Kaas (2014) consider a related model with default in equilibrium.

4

this model because the interest rate is always smaller than (1 + g)/β (see Propositions 1

and 2 below), and the absence of collateral prevents any borrowing. To keep the model

simple, I assume further that labor supply is constant, normalized to unity. Due to these

assumptions, workers’ consumption is simply cwt = wt +Xt.

Definition: Given deficit policy (Xt)t≥0, and given an initial distribution of capital, credit

and bond holdings (Ki0, B

i0, D

i0)i∈[0,1], a competitive equilibrium is a list of consumption,

investment, and production plans for all firms, (C it , K

it+1, B

it+1, D

it+1, L

it)i∈[0,1], t≥0, con-

sumption of workers, cwt = wt + Xt, t ≥ 0, public bonds (Dt+1)t≥0, and prices for labor,

credit and public bonds, (wt, Rt+1, qt)t≥0, such that

(i) (C it , K

it+1, B

it+1, D

it+1, L

it)t≥0 maximizes firm owner i’s expected discounted utility (1)

subject to credit constraints (2) and budget constraints (3).

(ii) Markets for labor, credit and public bonds clear. That is, in very period t ≥ 0,

∫ 1

0

Lit di = 1 ,

∫ 1

0

Bit+1 di = 0 ,

∫ 1

0

Dit+1 di = Dt+1 .

(iii) The government’s flow budget constraints (4) are satisfied for all t ≥ 0.

2.2 Equilibrium Characterization

Consider firm i which operates the capital stock Kit in period t at productivity zit. The

firm’s labor demand can be written as Lit = φ(wt/At)zitK

it/At, where the downward–sloping

function φ is the inverse of the marginal product of labor, i.e. F2(1, φ(ω)) = ω. Writing

Kt ≡

∫ 1

0

Kit di and Zt ≡

∫ 1

0

zitKit

Ktdi (5)

to measure aggregate capital and aggregate productivity, the labor market is in equilibrium

if

1 = φ(wtAt

)ZtKtAt

. (6)

This also implies that firm i’s profit before interest payments can be written as

Y it − wtL

it = f ′

(ZtKtAt

)

zitKit ,

with f(k) ≡ F (k, 1). Aggregating over the right-hand sides of budget constraints (3) yields

aggregate wealth

Wt ≡ f ′

(ZtKtAt

)

ZtKt + (1− δ)Kt + qtDt . (7)

5

Now consider the firm owners’ investment possibilities in period t−1. Each unit of capital

invested in firm i yields gross capital return f ′(ZtKt/At)zit + 1− δ in period t. Hence, the

firm owner will invest in his business if this return exceeds Rt; otherwise all funds will be

invested in financial assets. For the owner of an unproductive firm to be willing to supply

credit, the inequality Rt ≥ qt/qt−1 must hold.5 Furthermore, for government bonds to be

held by investors, the arbitrage condition

Rt =qtqt−1

(8)

must hold.

If f ′(ZtKt/At)zit + 1 − δ > Rt, firm owner i is borrowing constrained in period t − 1 and

borrows up to the leverage limit Bit/K

it = θt with

θt ≡λ(1− δ)

Rt. (9)

In this case, the owner’s return on equity investment Kit −Bi

t = (1− θt)Kit is

Rit ≡

f ′(ZtKt)zit + 1− δ − θtRt

1− θt, (10)

which exceeds the interest rate and the return on government bonds; hence, Dit = 0.

Conversely, if f ′(ZtKt/At)zit + 1 − δ ≤ Rt, the financial return Rt exceeds the capital

return, so that the owner will not invest in its own business (i.e. Kit = 0) if this inequality

is strict. The owner then supplies credit and possibly holds government bonds as well.

These considerations imply that there is a productivity cutoff zt defined by

Rt = f ′(ZtKtAt

)

zt + 1− δ , (11)

such that any owner’s return on savings is equal to Rit if z

it > zt and equal to Rt if z

it ≤ zt.

When Sit denotes savings, budget constraints (3) can be expressed as C it + Sit+1 = Ri

tSit

with Rit ∈ {Rt, R

it}. Because of logarithmic utility, any firm owner saves fraction β of

wealth and consumes the rest. Therefore, the aggregate equity that firm owners invest in

their own businesses in period t is βWt[1−G(zt+1)] so that the aggregate capital stock is

Kt+1 = βWt1−G(zt+1)

1− θt+1

. (12)

5If that inequality would fail, all wealth of the owners of unproductive firms would be invested in

government bonds, so that credit supply would be zero. This cannot be an equilibrium because credit

demand is positive since λ > 0.

6

Fraction θt+1 of the aggregate capital stock is financed externally, hence θt+1Kt+1 is credit

demand in period t. Aggregate credit supply is equal to aggregate savings of the owners of

less productive firms, βWtG(zt+1), less than the demand for government bonds.6 Therefore,

the credit market is in equilibrium if

βWtG(zt+1)− qtDt+1 = θt+1Kt+1 . (13)

Since idiosyncratic productivity draws are uncorrelated across time, zit is not correlated

with the equity, and thus with the capital stock Kit across firms. Hence, aggregate capital

productivity, as defined in (5), can be written

Zt =1

1−G(zt)

∫

zt

z dG(z) = E(z|z > zt) ≡ Z(zt) . (14)

With these considerations, a competitive equilibrium can be conveniently characterized as

a solution(

wt, Rt, zt, Zt, Kt,Wt, θt, qt, Dt

)

(15)

to the nine equations (4), (6)–(14).

Given an equilibrium, aggregate output is

Yt = f(ZtKtAt

)

At .

With a Cobb-Douglas production function f(k) = kα, for example, TFP is simply expressed

as Zαt A

1−αt . Although the previous exposition does not consider stochastic shocks, the term

Zt (and therefore TFP) can possibly fluctuate in response to exogenous changes in the

idiosyncratic productivity distributionG (e.g. uncertainty shocks), to variations in financial

conditions (modeled as shocks to parameter λ, for example), or to the government’s debt

policy. In all these cases, the cutoff productivity level zt shifts endogenously which directly

impacts TFP.

Since this is a growing economy, it helps to describe the dynamics in terms of a number of

stationary variables. For this purpose, define kt ≡ Kt/At, yt = Yt/At and dt = (qtDt)/At.

Further suppose that the government’s deficit policy is exogenously specified in terms of

the deficit-output ratio, denoted by ξt ≡ Xt/Yt = Xt/(Atyt).

In the Appendix (Lemma 1), I show that the model dynamics can be reduced to three

dynamic equations in the variables kt, zt and dt which, together with a given deficit policy

6Due to the assumption of a continuous productivity distribution, there is an atomistic mass of firms

whose productivity equals exactly zt+1. Those firm owners are indifferent between investing in their own

business or in financial assets, but their behavior does not affect the equilibrium.

7

(ξt)t≥0, describe a competitive equilibrium. To save notation, write Zt = Z(zt) for average

capital productivity and Rt = R(kt, zt) = ztf′(Z(zt)kt) + 1 − δ for the interest rate. The

first dynamic equation describes the equality between aggregate savings (right-hand side)

and aggregate investment in capital and in government bonds (left-hand side):

kt+1 +dt+1

Rt+1=

β1 + g

[

f ′(Ztkt)Ztkt + (1− δ)kt + dt

]

. (16)

The second equation follows from aggregate capital investment (12):

kt+1[Rt+1 − λ(1− δ)] =βRt+11 + g [1−G(zt+1)]

[

f ′(Ztkt)Ztkt + (1− δ)kt + dt

]

. (17)

The third equation is a reformulation of the government budget constraint (4):

ξtf(Ztkt) =1 + gRt+1

dt+1 − dt . (18)

3 Stationary Equilibria

This section describes all stationary equilibria (balanced growth paths) of this model for

stationary deficit policy ξt = ξ. It is convenient to specify steady states in terms of the

interest rate R = zf ′(Zk) + 1− δ and the productivity threshold z. Solving (16) and (17)

for the stationary debt-capital ratio gives

dk=RG(z)− λ(1− δ)

1−G(z). (19)

Substitution back into (17) gives rise to

(1 + g)[R− λ(1− δ)] = βR{

(1− δ)(1− λ) + (R− 1 + δ)[G(z) + (1−G(z))Z(z)z ]

}

. (20)

The capital stock in efficiency units k(z, R) follows from R = f ′(Z(z)k)z + 1 − δ. Then,

write the elasticity of the production function as α(z, R) = f ′(Z(z)k(z,R))Z(z)k(z,R)f(Z(z)k(z,R))

and solve

(18) fordk= R

1 + g − RZ(z)z (R− 1 + δ)

ξα(z, R)

. (21)

Equating to (19) yields

ξα(z, R)

[1−G(z)]Z(z)z (R + δ − 1) = [RG(z)− λ(1− δ)][

1 + gR − 1] . (22)

Any solution (z, R) to equations (20) and (22) specifies a steady state equilibrium, provided

that government debt is non-negative, which entails the restriction RG(z) ≥ λ(1− δ) (see

equation (17)).7

7A model solution with RG(z) < λ(1 − δ) would describe an implausible situation in which the gov-

ernment is a net creditor to the private sector, earning the same return R as private investors.

8

I start with the case of zero deficits (ξ = 0) so that the (nominal) stock of debt D is

constant. Due to the presence of credit constraints, this economy can permit an equilibrium

in which the government runs a Ponzi scheme, rolling over the existing stock of debt from

period to period. In such situations, the price of government debt is a bubble which has

positive value to investors because the rate of return equals the economy’s growth rate.

As shown below, the bubble equilibrium exists if and only if the economy with unvalued

outside assets is “dynamically inefficient”; that is, if the interest rate at the (unique) no-

bubble equilibrium is smaller than the growth rate. This feature is similar to well-known

results for overlapping-generations economies (Diamond (1965)) or for incomplete-markets

models (Aiyagari and McGrattan (1998)). Different from this literature, I show that the

bubble equilibrium has higher TFP than any equilibrium without a bubble.

A no-bubble stationary equilibrium has d = 0 which implies RG(z) = λ(1− δ). Then (20)

implies that the productivity threshold in a no-bubble equilibrium zN solves the equation

λ = G(z)[

1 +z

Z(z)

( 1 + g

β(1− δ)− 1

)]

. (23)

For existence and uniqueness, the following two assumptions on the productivity distribu-

tion will be required.

Assumption 1: The distribution function G has connected support suppG and is such

that z/Z(z) is non-decreasing for all z ∈ suppG.

Assumption 2: The distribution function G is such that z/Z(z) tends to one when z

converges to the supremum of suppG.

Assumption 1 is a relatively mild condition which is satisfied for uniform, exponential,

Pareto or log-normal distributions. Assumption 2 obviously holds for any bounded distri-

bution, but also for exponential or log-normal distributions.

Proposition 1: A no-bubble stationary equilibrium (zN , RN) exists if either Assumption

2 holds or if λ is sufficiently small. The interest rate satisfies RN = λ(1−δ)G(zN )

< β/(1 + g).

Under Assumption 1, a no-bubble stationary equilibrium is unique.

The possibility of a bubble stationary equilibrium arises in those cases where the no-bubble

equilibrium is dynamically inefficient. In a bubble stationary equilibrium, the price of the

fixed stock of debt D grows at rate g, so that R = 1 + g. It follows from (20) that the

9

productivity threshold zB in a bubble equilibrium solves

1− β

β[1 + g − λ(1− δ)] = (1−G(z))

[Z(z)

z− 1

]

(g + δ) . (24)

A solution to this equation is a bubble equilibrium only if G(zB) > λ(1− δ)/(1+ g) which

implies that public bonds have strictly positive value.

Proposition 2: Suppose that Assumption 1 holds. Then a unique bubble stationary equi-

librium (zB , RB) with RB = 1 + g exists if and only if RN < 1 + g. Moreover, zB > zN

holds so that TFP is higher in the bubble stationary equilibrium.

The existence of a bubble stationary equilibrium depends on the tightness of credit con-

straints. With tighter constraints, lower demand for credit reduces the interest rate so that

dynamic inefficiency (overaccumulation of capital) can arise which is a prerequisite for a

bubble equilibrium. While the bubble asset (here unbacked government debt) crowds out

capital, it has a positive impact on factor productivity via the capital allocation among

heterogeneous firms. This is because some low-productivity firms decide to stay out of

business at the higher interest rate so that capital is employed more productively. While

the ultimate effect of the bubble on output seems ambiguous, I show in the calibrated

example of the next section that output is higher at the bubble equilibrium. That is, the

increase in TFP more than offsets the decline in the aggregate capital stock.

Regardless of the existence of a bubble, one can show that a relaxation of credit constraints

has an unambiguous positive effect on TFP. In the absence of a bubble, this is because low-

productivity firms find it less valuable to borrow when the interest rate rises in response

to a credit expansion. If the economy exhibits a bubble, a credit expansion does not

affect the interest rate (which equals the growth rate), yet it also has a positive effect on

factor productivity: higher demand for credit leads to more capital investment and to less

investment in government bonds. This reduces the capital return for all firms which makes

it again less valuable for low-productivity firms to borrow and to invest. Consequently,

capital is more efficiently allocated and TFP rises. These assertions are summarized as

follows:

Proposition 3: A credit expansion induced by an increase of parameter λ raises zN and

zB, and therefore total factor productivity at any stationary equilibrium.

While Proposition 2 establishes the precise condition under which unbacked government

debt can be rolled over indefinitely, the very same condition also permits the perpetual

10

financing of primary deficits, as long as these are not too large. To obtain an analytical

result, it simplifies to consider a Cobb-Douglas production function so that the elasticity

α(z, R) is constant.

Proposition 4: Suppose that the production function is f(Zk) = (Zk)α. Under Assump-

tion 1 and RN < 1 + g, there is a maximum sustainable deficit ξ > 0 such that any

economy with deficit-output ratio ξ ∈ [0, ξ) has two stationary equilibria (zi, Ri)i=1,2 with

zN ≤ z1 < z2 ≤ zB and RN ≤ R1 < R2 ≤ 1 + g.

This proposition extends Proposition 2 to positive deficit levels. The low-productivity

equilibrium (z1, R1) is a continuation of the no-bubble equilibrium which is identical to

(zN , RN) when ξ = 0. Conversely, the high-productivity equilibrium (z2, R2) coincides

with the bubble equilibrium when ξ = 0. Consequently, it is appropriate to label (z1, R1)

a no-bubble steady state and (z2, R2) a bubble steady state. Both equilibria collapse when

the primary deficit reaches the maximum sustainable level ξ. Values of the deficit above

ξ are not sustainable, i.e. they lead to exploding debt stocks and hence do not permit a

stationary equilibrium.

Without providing a formal stability analysis for the three-dimensional dynamic system

(16)–(18) in (kt, zt, dt), it is worth noting that the low-productivity (no bubble) equilibrium

at z1 is typically a sink while the bubble equilibrium is a saddle, which is similar to related

findings for overlapping-generations models with unbacked government debt (cf. Chalk

(2000)). If the bond price qt (and hence dt) is allowed to jump in response to contem-

poraneous shocks, the bubble equilibrium should be regarded as a (locally) determinate

equilibrium (i.e., there is no other equilibrium nearby the stationary equilibrium) while

the no-bubble equilibrium is locally indeterminate: that is, bond prices may be subject to

sunspot fluctuations, so that an indeterminate equilibrium dynamics arises around the sta-

tionary equilibrium at (k1, z1, d1). Without such instantaneous adjustments of bond prices

(e.g. if bond yields were indexed so that government debt pays always the same return as

private credit), the bubble equilibrium should be regarded as an unstable state, while the

no-bubble equilibrium is locally stable (see e.g. Chalk (2000)). While both interpretations

are principally acceptable, this paper favors the second view and treats dt as a predeter-

mined variable. This can be justified by the observation that empirical bond returns are

rather stable and do not exhibit the large fluctuations (relative to capital returns) that

would be required to keep the economy at the determinate saddle path in response to

fundamental economic shocks.

11

4 Quantitative Implications

This section describes how the model is calibrated to match certain long-run features for

the US economy. Subsequently the calibrated model is used to address several quantitative

questions: First, what is the maximum sustainable level of the primary deficit? Similarly,

what is the maximum sustainable stock of public debt, given the calibrated value of the

primary deficit? Second, what are the consequences of temporary or permanent changes

in fiscal deficits on TFP and output? Third, how does the economy respond to a shock to

credit market conditions?

4.1 Calibration

I calibrate the model at annual frequency and set parameters to match selected data targets

for the period 1970–2013. Regarding the technology, I choose a Cobb-Douglas production

function y = (Zk)α and set α = 0.3 to match a labor income share of 70 percent. The

growth rate of labor efficiency is set to g = 0.028 to match the average growth rate of US

real GDP (output). The depreciation rate δ is set to the data average of 5.4 percent and

the discount factor β is set to match a capital-output ratio of 2.68.8

For the idiosyncratic capital productivity distribution, it is convenient to use a Pareto

distribution of the form G(z) = 1− (z0/z)ϕ with scale parameter z0 and shape parameter

ϕ > 1. As the scale parameter can be arbitrarily normalized, I set it to the value such that

TFP (i.e., Zα) equals unity in steady state. Because of Z(z) = E(z′|z′ ≥ z) =ϕzϕ− 1, the

gross interest rate can be written

R = zα(kZ)α−1 + 1− δ = zZα

(Zk)α

k+ 1− δ =

ϕ− 1ϕ αYK + 1− δ . (25)

To obtain a data target for the real interest rate (i.e., the real return on government

bonds), I calculate the difference between the average annualized 3-months treasury bill

rate and the average growth rate of the GDP deflator. This gives R = 1.016 which implies

ϕ = 2.56. Because this shape parameter is crucial for the following quantitative findings,

it is appropriate to compare its implications for other data moments. First, note that

the gap between Z and z not only pins down the safe interest rate but also the average

equity premium in this model, which is the difference between the average equity return

[f ′(Zk)Z + 1 − δ − θR]/(1 − θ) (see equation (10)) and the safe interest rate R. For the

calibrated parameter values, the equity premium attains the reasonable value of 5.4%.

8The capital stock is calculated from real net domestic investment using the perpetual inventory method

based on the assumption that the capital-output ratio is constant.

12

Second, parameter ϕ has direct implications for the cross-sectional productivity disper-

sion. Since ln(z) is exponentially distributed with parameter ϕ, the standard deviation of

ln(TFP) = α ln(z) across active firms is α/ϕ which is 0.12 for the calibrated parameter

values. If firms are weighted by size (note that employment and output are proportional

to the idiosyncratic draw of zi), the weighted standard deviation of ln(TFP) turns out to

be α/(ϕ − 1) ≈ 0.19. Both measures are smaller than the empirical measures of stan-

dard deviations of ln(TFP) for US firms within narrowly defined industries which average

around 0.4 (see Hsieh and Klenow (2009) and Bartelsman et al. (2013)). While model

calibrations with more dispersion (i.e. lower ϕ) are principally feasible, they would require

either a negative real interest rate or implausibly low values of the capital-output ratio,

see equation (25). For instance, with ϕ = 1.75 (and the same calibration target for K/Y ),

the employment-weighted standard deviation of ln(TFP) would increase to 0.4, but the

real interest rate would fall to −0.6% and the equity premium would increase to 8.2%.

The collateral share parameter λ is calibrated so that the volume of business credit is

58 percent of annual output, while the primary deficit ratio ξ is calibrated to match a

public-debt-output ratio of 50 percent.9 The calibrated parameter value for the (primary)

deficit-output ratio ξ is 0.59%, which slightly exceeds the data average10 of 0.31%. All

parameter choices are summarized in Table 1. It is worth mentioning that the calibration

targets are hit exactly and pin down the model parameters uniquely (see the Appendix).

4.2 Results

To determine the maximum sustainable deficit for this calibration, Figure 1 shows how the

steady states depend on the deficit-output ratio (parameter ξ). The solid curves in these

graphs show the no bubble steady states which feature lower TFP than the bubble steady

states shown by the dashed curves (cf. Proposition 4). At the same time, no-bubble steady

states exhibit lower interest rates, more private credit and hence a larger capital stock,

which is similar to dynamically inefficient overlapping-generations production economies

(Diamond (1965)). But while in those economies the no-bubble equilibrium has higher

output, the opposite is true in this model: despite a higher capital stock, output is smaller

at the no-bubble equilibrium compared with the bubble equilibrium. This is so because of

9To obtain these data targets (averages over 1970–2013), I use credit market liabilities of the non-

financial business sector and credit market liabilities of the general government (consolidated) from the

Flow of Funds Accounts of the Federal Reserve (Tables L.101 and L.105.c).10The deficit-output ratio is calculated as net borrowing minus interest payments of the general govern-

ment (consolidated) from the Flow of Funds Accounts (Table F.105.c), divided by nominal GDP.

13

Table 1: Parameter choices.

Parameter Value Description Target

δ 0.054 Depreciation rate Consumption of fixed capital

α 0.3 Production fct. elasticity Capital income share

g 0.028 Labor efficiency growth GDP growth

β 0.978 Discount factor Capital-output ratio

ϕ 2.67 Pareto shape Real interest rate

z0 0.535 Pareto scale Normalization Z = 1

ξ 0.0059 Deficit-output ratio Public-debt-output ratio

λ 0.236 Collateral share Firm-credit-output ratio

lower TFP: at low interest rates more low-productivity firms are active which leads to a

less efficient capital allocation.

The dot in Figure 1 shows the calibrated steady state which is a no-bubble equilibrium

and therefore stable. Note that this feature is not imposed in the calibration but is a result

of the relatively large difference between the average output growth (g = 2.8%) and the

safe interest rate (R − 1 = 1.6%). A higher target for the interest rate would imply that

the calibrated steady state is a bubble (and hence unstable).

As can also be seen from Figure 1, the maximum sustainable deficit ratio implied by

this calibration is at ξ = 0.834%. This is much lower than the value for the maximum

deficit obtained from quantitative overlapping generations models: Chalk (2000) finds that

primary deficits up to 5.2% are sustainable, while Bullard and Russell (1999) calibrate a

similar model with a primary deficit of 1.9%. Both papers have an even larger gap between

the growth rate and the real interest rate. In the model of this paper, raising the deficit to

the maximum sustainable level would increase the public-debt-output ratio to 114% while

output increases slightly, which is due to higher TFP albeit at a reduced capital stock.

A related policy issue is to determine the maximum sustainable debt stock under the

presumption that the government implements the actual (calibrated) primary deficit ratio

ξ = 0.59%. Simulations reveal that debt stocks above 203% of output lead to explosive

dynamics whereas any lower debt level is sustainable under the current deficit policy.

While these considerations compare steady state outcomes, it is also instructive to explore

the transition dynamics under a change of policy. Figure 2 considers a scenario in which

the government starts to implement a zero-deficit policy from year t = 1 onwards, thus

reducing the debt-output ratio to zero in the very long run. Indeed, it takes quite some

14

Figure 1: Steady state values for varying deficit-output ratio ξ. The dot shows the cali-

brated steady state. Solid curves show no-bubble (sink) steady states and dashed curves

show bubble (saddle) steady states.

time to deplete the debt stock: after 50 (100; 200) years, the debt-output ratio falls to

25.5% (12.3%; 2.7%). As shown by the intersection between the solid curves and the

vertical axes in Figure 1, the zero-deficit steady state has a higher capital stock, lower

TFP and slightly lower output. However, the transition dynamics in Figure 2 reveals that

it takes a very long time (almost 140 years) until (detrended) output falls below the initial

level. This is because the private credit and investment boom induced by the public debt

reduction dominates output dynamics initially, while the TFP reductions operate much

later and are quantitatively less important in the first century after the policy change.

A more topical policy experiment concerns the strong increase of the public debt stock

induced by expansive fiscal policy in the aftermath of the Great Recession. To study the

effects of this policy, Figure 3 shows a scenario in which the government raises the primary

15

Figure 2: Adjustment dynamics to a zero-debt steady state (zero primary deficit from

t = 1 onwards). All variables are expressed as percentage deviations from their steady

state values.

deficit ratio to ξ = 6% over five years (the average value over the period 2009-2013) before

it returns to the stationary (calibrated) deficit policy. As shown by the lower graphs in

the figure, public debt increases by 54% which substantially crowds out private credit and

capital. It also takes a long time to reduce the public debt stock: after 50 years, debt

is still 45% above the initial level. Yet, the upside of the deficit policy is an increase of

TFP by about 1% which dampens the initial decline in output, although the quantitative

effect of the crowding out of capital dominates output dynamics, similar to the previous

findings. After three decades, however, detrended output slightly exceeds the level before

the policy change, until it converges back to the initial level.

Since this economy features productivity losses because of binding credit constraints, it is

also interesting to consider the response to a change in credit market conditions (“financial

shock”). Figure 4 shows the model adjustment to a permanent 10% increase of the collat-

eral parameter λ, capturing the effect of a credit expansion which is similar in magnitude

to the period 1995–2005. Relaxed borrowing constraints have a twofold (albeit quantita-

tively small) effect on output. First, TFP increases by about 0.3% which is induced by a

16

Figure 3: Adjustment dynamics with primary deficit ξt = 6% for t = 1, . . . , 5 and ξt =

0.0059 for t ≥ 6. All variables are expressed as percentage deviations from their steady

state values.

small increase of the interest rate. Over time, the capital stock increases by 0.45% which

reduces the capital return and the interest rate, leaving TFP unaffected. The combined

long-term effect on output is just below half a percentage point. Hence in this economy

the quantitative effect of credit market conditions on output is quite modest.

5 Conclusions

This paper presents a macroeconomic model that includes two key ingredients: (i) endoge-

nous TFP coming from financially constrained firms operating at different productivity

levels; (ii) the possibility of sustainable primary deficits that arises from low equilibrium

interest rates. The second feature is closely linked to the first one: if financial constraints

are sufficiently tight, the interest rate falls below the growth rate in which case the govern-

ment is able to role over unbacked public debt indefinitely. Those dynamically inefficient

economies typically feature two steady states: one unstable “bubble” steady state with a

higher interest rate, higher public debt and a lower capital stock, and a stable “no bubble”

17

Figure 4: Adjustment dynamics to a permanent financial shock (λt = 0.26 from t = 1

onwards). All variables are expressed as percentage deviations from their steady state

values.

steady state with a lower interest rate, less public debt and more private capital. While

these features are also obtained in overlapping generations models, the endogenous TFP

of this model reverses the implications for aggregate output: the bubble steady state has

higher TFP and output (despite a lower capital stock) than the no-bubble steady state.

When calibrated to the US economy, the model is used to study a few quantitative ques-

tions. I find that the maximum sustainable primary deficit ratio is less than one per-

cent, and hence much smaller than related numbers from quantitative studies based on

overlapping-generations models. Moreover, the endogenous responses of TFP to changes

in debt policy are quantitatively modest and are typically dominated (at least in the

medium run) by the crowding out of private credit and capital.

An attractive feature of this model is that it is quite tractable, described by only few state

variables and dynamic equations, which makes it rather flexible for extensions. It should

be straightforward to augment the model by adding flexible labor supply and by fleshing

out more detailed fiscal policy instruments. Such extensions would make the model more

interesting for business-cycle analysis in order to quantify the importance of various shocks,

18

including uncertainty shocks or financial shocks. Model extensions could also be used to

study the impact of fiscal rules on the dynamics of output, credit and TFP.

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20

Appendix

Lemma 1: A competitive equilibrium is described by solutions in (zt, kt, dt) to the dynamic

system defined by (16), (17) and (18).

Proof: Rewrite equations (7), (12) and (13) in terms of kt = Kt/At, ωt = Wt/At, dt =

(qtDt)/At:

ωt = f ′(ktZt)Ztkt + (1− δ)kt + dt , (26)

kt+1 = βωtRt+1(1−G(zt+1))

(1 + g)[Rt+1 − λ(1− δ)], (27)

βRt+11 + g ωtG(zt+1) = λ(1− δ)kt+1 + dt+1 . (28)

Substitution of (26) into (27) directly yields (17). Combine (27) and (28) to obtain

kt+1[Rt+1 − λ(1− δ)] = β ωt1 + g [1−G(zt+1]Rt+1

= β ωt1 + gRt+1 − dt+1 − λ(1− δ)kt+1 .

Substitution of (26) yields (16). With Xt/At = ξtyt = ξtf(Ztkt), the government budget

constraint (4) is

ξtf(Ztkt) =(1 + g)dt+1

Rt+1− dt ,

which is equation (18). ✷

Proposition 1: A no-bubble stationary equilibrium (zN , RN) exists if either Assumption

2 holds or if λ is sufficiently small. The interest rate satisfies RN = λ(1−δ)G(zN )

< β/(1 + g).

Under Assumption 1, a no-bubble stationary equilibrium is unique.

Proof: Let z ≥ 0 and z > 0 denote the upper and lower bounds of the support of G, with

z = ∞ if G has unbounded support. The right-hand side (RHS) of (23) is zero at z = z and

positive at higher values of z. Hence a solution exists if λ is sufficiently small. Moreover,

under Assumption 2 the RHS tends to (1 + g)/(β(1− δ)) > 1 for z → z, so that (23) has

a solution for any λ ≤ 1. Because of z/Z(z) < 1, it follows that λ < G(zN )1 + g

β(1− δ)and

hence RN =λ(1− δ)G(zN )

<1 + gβ

.

Under Assumption 1, the RHS is strictly monotonically increasing. Hence, there can be

at most one solution to equation (23). ✷

21

Proposition 2: Suppose that Assumption 1 holds. Then a bubble stationary equilibrium

exists if and only if RN < 1 + g. Moreover, zB > zN holds so that TFP is higher in the

bubble stationary equilibrium.

Proof: Let z be the unique productivity value where G(z) = λ(1 − δ)/(1 + g) < 1. A

stationary bubble equilibrium is a solution zB of (24) such that zB > z. Write φ(z) for the

RHS of (24). Since φ is strictly decreasing and tends to zero for z → z, a bubble stationary

equilibrium exists if and only if the LHS of (24) is smaller than φ(z) which is equivalent to

(1 + g)1− β

β<

[Z(z)

z− 1

]

(g + δ) . (29)

Now suppose that RN < 1 + g. Because of

RN =λ(1− δ)

G(zN )= 1− δ +

zN

Z(zN )

(1 + g

β− 1 + δ

)

< 1 + g ,

it follows that zN > z, and therefore

1 + g

β− 1 + δ <

Z(zN )

zN(g + δ) ≤

Z(z)

z(g + δ) ,

where the last inequality follows since Z(z)/z is non-increasing. This proves the existence

of a bubble stationary equilibrium.

Conversely, suppose that

RN =λ(1− δ)

G(zN )= 1− δ +

zN

Z(zN )

(1 + g

β− 1 + δ

)

≥ 1 + g .

This implies that zN ≤ z and therefore

1 + g

β− 1 + δ ≥

Z(zN )

zN(g + δ) ≥

Z(z)

z(g + δ) .

This proves that no bubble stationary equilibrium exists if RN ≥ 1 + g.

Finally, it remains to show zB > zN whenever a bubble stationary equilibrium exists. This

holds iff1− β

β[1 + g − λ(1− δ)] = φ(zB) < φ(zN ) . (30)

Because of

RN =λ(1− δ)

G(zN )= 1− δ +

zN

Z(zN )

(1 + g

β− 1 + δ

)

22

one has

φ(zN ) = [1−G(zN )][Z(zN )

zN−1

]

(g+ δ) = [1−G(zN )]g + δ

β(1− δ)

G(zN )(1 + g)− β(1− δ)λ

λ−G(zN).

To show (30), it must be proven that

1− β

β[1 + g − λ(1− δ)] < [1−G]

g + δ

β(1− δ)

G(1 + g)− β(1− δ)λ

λ−G︸ ︷︷ ︸

≡ψ(G)

(31)

holds at G = G(zN ). Because a bubble equilibrium exists, RN < 1 + g implies that

G(zN ) > λ(1 − δ)/(1 + g), and (23) implies that G(zN ) < λ. It follows ψ(λ(1 − δ)/(1 +

g)) = (1/β − 1)[1 + g − λ(1 − δ)] and ψ(λ) = ∞. Therefore (31) holds if ψ′(G) > 0 for

G ∈ [λ(1− δ)/(1 + g), λ) which is easy to verify. This proves that zB > zN . ✷

Proposition 3: Under Assumption 1, a credit expansion induced by an increase of parame-

ter λ raises zN and zB, and therefore total factor productivity at any stationary equilibrium.

Proof: Consider first the no-bubble stationary equilibrium at zN . Because the RHS in (23)

strictly increases in z under Assumption 1, zN (and hence TFP which depends positively

on Z(z)) increases if λ increases. Second consider a bubble stationary equilibrium at

zB. Again Assumption 1 implies that the RHS of (24) decreases in z, and since the LHS

decreases in λ, zB must strictly increase if λ increases. ✷

Proposition 4: Suppose that the production function is f(Zk) = (Zk)α. Under Assump-

tion 1 and RN < 1+g, there is a maximum sustainable deficit ξ > 0 such that any economy

with deficit-output ratio ξ ∈ [0, ξ) has two stationary equilibria (zi, Ri)i=1,2 with z1 < z2

and R1 < R2 < 1 + g.

Proof: A steady state equilibrium (R, z) is a solution to equations (20) and (22). For any

given z, the RHS of (20) is a quadratic function of R which intersects the LHS uniquely at

some R > 1−δ; see Figure 5. This follows because LHS(R = 1−δ) > RHS(R = 1−δ) > 0

and RHS(R = 0) = 0, so that the RHS is quadratic in R and strictly increasing in R > 1−δ.

Since the RHS is strictly decreasing in z, equation (20) defines an upward-sloping solution

R(z) ≥ 1− δ such that R(z) → (1 + g)/β when z → z.

It will now be shown that R(z) intersects twice with the curve that defines implicit solutions

(R, z) to the other equilibrium condition (22) for small enough values of the deficit ratio

ξ. Because two solutions at z = zN and at z = zB exist for ξ = 0, the implicit function

23

)1( dl -0

LHS of (20)

RHS of (20)

R)(zR

Figure 5: Equation (20) defines R(z).

theorem also guarantees two (locally unique) solutions for small values of ξ, provided that

the two curves (20) and (22) have different slopes at zB and at zN for ξ = 0. But this is

easy to verify. At z = zB and ξ = 0, (22) defines the flat curve R = 1 + g, whereas at

z = zN , equation (22) defines the downward-sloping R = λ(1− δ)/G(z); see Figure 6. As

shown above, (20) defines an upward-sloping curve R(z). Therefore, for any small enough

ξ > 0, the two equations have two solutions (zi, Ri), i = 1, 2 such that z1 → zN and

z2 → zB when ξ → 0.

It still remains to verify that public debt is non-negative at either solution. This follows

trivially by continuity for the solution z2 because of RBG(zB) > λ(1 − δ). For the other

solution (z1, R1), rewrite equation (22) as

φ(z, R, ξ) ≡ [RG(z)− λ(1− δ)][1 + g −R]−ξα(1−G(z))R(R + δ − 1)

Z(z)z .

The local derivatives at (z, R, ξ) = (zN , RN , 0) have signs

φ1 = G′(zN )RN(1 + g −RN ) +ξαR

N (RN + δ − 1)(zN)2G′(zN ) +

∫

zNz dG(z)

(zN )2> 0 ,

and φ3 < 0 which implies that dzdξ

= −φ3/φ1 > 0 at any given R. Hence, the curve (22) lies

above the curve R = λ(1 − δ)/G(z) in (z, R) space for ξ > 0, which implies that z1 > zN

and R1 > RN , and hence R1G(z1) > λ(1− δ), so that public debt is positive.

24

Equation (22)

Equation (20), ( )R z

zN

z

R

g+1

Bz1z

2z

)(

)1(

zGR

dl -=

Figure 6: Intersections between equations (20) and (22) define z1 and z2.

Lastly, because both solutions (zi, Ri), i = 1, 2, satisfy the upward-sloping relation R =

R(z), the equilibrium with the higher interest rate has higher TFP. ✷

Calibration

Parameters α, δ and g are directly calibrated as mentioned in the text. Given the calibra-

tion targets for the interest rate R and for the debt-output ratio D/Y , the deficit-output

ratio follows from ξ = (D/Y ) · (g − R)/R. The calibration target for K/Y together with

the normalization Z = 1 yields the steady-state value of k = K/A = (K/Y )1/(1−α). This

together with R = zf ′(Zk)+1−δ yields the steady-state value of z = (K/Y )·(R+δ−1)/α,

so that the Pareto shape parameter follows from ϕ = 1/(1− z). The calibration target for

the firm-credit-output ratio Credit/Y = θK/Y = λ(1− δ)/R · (K/Y ) identifies parameter

λ = (Credit/Y ) · R/(1 − δ)/(K/Y ). Then steady-state equation (22) can be solved for

G(z):

G(z) =

ξαz (R + δ − 1) + λ(1− δ)

g − RR

g − R +ξαz (R + δ − 1)

.

With G(z) = 1 − (z0/z)ϕ, this yields the Pareto scale parameter z0. Finally, the discount

factor follows uniquely from steady-state equation (20).

25

Un ive rs i ty o f K on s ta n z De pa rtmen t o f Econ om ics

UNIVERSITY OF KONSTANZ Department of Economics Universitätsstraße 10 78464 Konstanz Germany Phone: +49 (0) 7531-88-0 Fax: +49 (0) 7531-88-3688 www.wiwi.uni-konstanz.de/econdoc/working-paper-series/